If `m+ (1)/(m+2)= 0` find
`m^(4) + m^(3) + m^(2) + m+1`=?

Answer

To solve the equation \( m + \frac{1}{m + 2} = 0 \) and find the value of \( m^4 + m^3 + m^2 + m + 1 \), we will follow these steps:

### Step 1: Solve for \( m \)

Starting with the equation:
\[ m + \frac{1}{m + 2} = 0 \]

We can rearrange this equation:
\[ m = -\frac{1}{m + 2} \]

### Step 2: Eliminate the fraction

To eliminate the fraction, multiply both sides by \( m + 2 \):
\[ m(m + 2) = -1 \]

This expands to:
\[ m^2 + 2m + 1 = 0 \]

### Step 3: Factor the quadratic equation

The equation can be factored as:
\[ (m + 1)^2 = 0 \]

### Step 4: Solve for \( m \)

Setting the factored equation to zero gives:
\[ m + 1 = 0 \implies m = -1 \]

### Step 5: Substitute \( m \) into the expression

Now we need to find \( m^4 + m^3 + m^2 + m + 1 \) with \( m = -1 \):
\[ (-1)^4 + (-1)^3 + (-1)^2 + (-1) + 1 \]

Calculating each term:
\[ 1 - 1 + 1 - 1 + 1 \]

### Step 6: Simplify the expression

Now, simplify:
\[ 1 - 1 + 1 - 1 + 1 = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]

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